Monte Carlo Integration

This package provides multidimensional integration algorithms based on monte carlo methods. The biggest advantage of using monte carlo methods is that their convergence rate is independent of the dimension of the integral.

Currently, this package only provides a routine called VEGAS:

vegas(f, st, en, kwargs...)

VEGAS is a Monte Carlo algorithm for multidimensional integration based on adaptive importance sampling. It divides each dimension into bins and adaptively adjusts bin widths so points are sampled from the region where the function has highest magnitude.

• st: Array of starting values in each dimension. Defaults to zeros(2)
• end: Array of ending values in each dimension. Defaults to ones(2)

• nbins: Number of bins in each dimension. Defaults to 100.
• ncalls: Number of function calls per iteration. Defaults to 1000.
• maxiter: Maximum number of iterations. Defaults to 100.
• rtol: Relative tolerance required. Defaults to 1e-4.
• atol: Absolute tolerance required. Defaults to 1e-2.
• debug: Prints abs(sd/I) every 100 iterations. Defaults to false.
• batch: Whether f returns batches of function evaluations. f is assumed to take one argument pts, an ncalls × ndims matrix. Each row is a unique point and returns an ncalls length vector of function evals. This argument defaults to false.

• Estimate for the integral
• Standard deviation
• χ^2 / (numiter - 1): should be less than 1 otherwise integral estimate should not be trusted.

• Lepage, G. Peter. "A new algorithm for adaptive multidimensional integration." Journal of Computational Physics 27.2 (1978): 192-203.

Most of the computation time in an integration algorithm is usually spent in function evaluations. The batch inteface allows users to provide batches of function evaluations, instead of supplying a function directly to be integrated. Users can now evaluate a number of points in parallel.

• Supporting vector valued functions
• Other integration algorithms